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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 45

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Answer: \frac{1}{2}\left ( \log x \right )^{2}+c

Hint: You must know about the integral rule of logarithmic functions.

Given: \int \frac{\log x}{x}dx


\int \frac{\log x}{x}dx

Put  \log x =t     and differentiate both sides,\frac{d}{dx}\log x= \frac{1}{x}

\begin{aligned} &\int \frac{\log x}{x} d x=\int t d t \\ &=\frac{t^{2}}{2}+c \\ &\frac{1}{x} d x=d t=\frac{(\log x)^{2}}{2}+c\quad\quad\quad\quad\quad\quad\quad\quad\left[\int \frac{1}{x} d x=\log x+c\right] \end{aligned}

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